Moment of Inertia and Frictional Torque

Objective:

• Determining the moment of inertia of a rotating disk and angular deceleration under the effect of friction torque.
• Calculating the time the cart takes to go down an inclined plane.

Physical thinking and derivation:

• We are asked to find the time the cart needs to go down 1m on an inclined plane theoretically, the most reasonable approach for this problem is to use torque clockwise is equal torque counterclockwise, force applied vertically up equals force applied vertically down, and force applied horizontally left equals force applied horizontally right. 
• In order to use those three equations above, we need to determine the moment of inertia of the rotating disk. 
• The rotating disk includes three parts with a large metal disk in the center, and two small shafts on each side. The large metal disk is cylindrical shaper, so do these two shafts on the side, but they are not symmetrical. An equation to calculate moment of inertia of the cylinder is half of the product of mass and radius squared. (Equation 1)

• Therefore, in order to calculate moment of inertia, we need to measure the radius of each disk and figure out their masses. We use a big caliper to measure the diameter of the large disk, and use small caliper to measure the diameters of two shafts on the side.
• Since three disks are connected to each other as a system, we can't actually measure the mass of each one respectively. The approach to solve the mass of each disk is to find the percent volume of each one, and multiply it with the total mass of the system. The volume of each one can be calculated using an equation below.

• Since we find the mass of a large disk and two small shafts on each side, we can use equation (1) to find the moment of inertia of each one, then add them up to get the moment of inertia of the whole system.
• Below is our calculation

Figure 1: Recording dimensions and calculating the percent volume of the large disk and two shafts.

Figure 2: Calculating the mass, moment of inertia of each disk, then find the moment of inertia of the whole system.

Calculating the angular deceleration due to the friction torque:

• Before we actually do the experiment with the cart, we run a first trial without the cart to figure out how we will calculate the angular acceleration. The reasons we need angular acceleration of rotating disk itself because we take the friction torque into account. Friction makes rotating disk slow down until it stops. Also, the friction torque is product of moment of inertia and angular acceleration. Therefore, we need to figure out angular acceleration in order to find friction torque (moment of inertia is already calculated before).
• From the 2-D collision lab, we know that capturing and analyzing video of two objects collide helps us obtain the value of linear position and velocity. We will use the same idea for this experiment. We will capture and analyze the video of the rotating disk rotates and stops by itself. The only force applied to rotating disk to make it stop is friction. Thus, we can figure out the angular acceleration of friction torque.
• Since analyzing the video, we will get the linear position and velocity. From graph of linear velocity vs. time, we can obtain the value of linear acceleration. Linear acceleration is related to angular acceleration through the following equation.

• To get the value of angular acceleration, we divide linear acceleration by the radius of the rotating disk.

Setting up the apparatus to obtain the angular acceleration.

• Let the rotating disk sit on the table, make sure that the table is leveled.
• Setting up the camera such that the video capturer parallels to the center of the rotating disk. Adjust the camera setting until we are satisfied with the quality. Connect the camera to the Logger Pro.
• We also mark a tape on the rotating disk to make it easy for analyzing it later.
• Give the rotating disk a gentle spin and let it go until it stops. Start capturing the video and analyze it later.

Below is the apparatus:

Figure 3: The apparatus of the experiment.

Data collection and analysis.

• Since we have the video, we analyze it by dotting the tape until it stops, setting scale and origin. The scale is the radius of the large disk, which is 0.09818m.

• Every dot on the disk gives us a set of values in Logger Pro, and here is the data table of linear position and velocity.

Figure 4: Data table of linear position and velocity

• Since the linear velocity is breaking down into two components x and y, we need to find the resulting velocity by taking square root of x-componet velocity squared and y-component velocity squared.

• Then graphing the resultant velocity vs. time, and linearizing the graph to obtain the linear acceleration.

Figure 5: Linearizing the graph of linear velocity vs. time.

• Finally, divide the linear acceleration by the radius of rotating disk to get the value of angular acceleration. Here is the work.

• Wow! Now we can use that value to calculate the friction torque which is our main goal from the beginning.

Actually lab:

• The main objective of this lab is to compare the theoretical and experiment value of time needed for a cart to roll down an inclined track for a distance of 1 meter. 
• We can obtain the experimental value of time by using the stopwatch on our phone. For theoretical value, we will obtain through our calculation.

Calculating the experimental value of time.

• To find the experimental value of time, we apply kinematic equation since the acceleration is constant.

• As the cart starts at rest, we don't worry about initial velocity. Therefore, all we need is to find the linear acceleration. The most reasonable approach is to acceleration is through the second Newton Law, which is F = m*a
• We know the mass of the cart, thus we just need to analyze those forces acting on the cart. There are three major forces: the gravity, the normal force, and tension. Since the cart is on the inclined plane, the gravity is broken down into two component x and y. Adding those forces in the direction of the cart (moving down the inclined plane), we have:

• We also notice that the tension is still unknown. Therefore, we will apply the definition of torque to figure out the tension, then use that value along with other known values to calculate the acceleration. 
• Substituting the value of acceleration back into equation (*) to find the theoretical value of time.
• Calculation shown in the figure below.

Figure 6: Finding the experimental value of time.

t=7.91s

Setting up an actual apparatus of the lab.

• Let an inclined track rest at the edge of the table. The inclined track makes with the floor a certain angle needed to measure.
• Connect a 500-gram dynamics cart to the rotating disk through a string. Make sure that the string is parallel to the ramp. 
• When everything is set up, let the cart roll down a distance of 1 meter and uses a stopwatch to determine the actual time. 

Figure 7: The apparatus of the experiment.

Discussion:

• From calculation, the time the cart will take to roll down a distance of 1 meter is 7.91m/s. For the actual time we obtain from experiment is 8.09 m/s. The percent error is calculated below. 

• We can calculate the uncertainty of this experiment, then compare it to the percent error to see how well we run the experiment. For the angle, we can check the uncertainty if we are off 1 degree. We have sin(49.6) = 0.762 and sin(50.6) = 0.773. The uncertainty of the angle is equal ((0.773 - 0.762) / 0.773 * 100%) = 1.4%. While the radius of the small disk number 1 give us the uncertainty (0.01 / 3.14 * 100%) = 0.3%. Additional, the uncertainty in moment of inertia is equal the sum of uncertainties from radii (0.3% + 0.3% + 0.05% = 0.65%. Thus, the total uncertainty is 1.4% + 0.3% + 0.65% = 2.35%. Moreover, we can add some more uncertainty from dotting the points when analyzing the video or timing the actual time when the cart rolls down a distance of 1 meter. This refers that the actual uncertainty of this experiment is greater than 2.35%. However, our percent error already falls within this region, which means we runt the experiment pretty well. 

Conclusion:

• The main goal of this lab is to compare the theoretical and experimental value of time when the cart rolls down a distance of 1 meter. However, we break down the lab into three small sessions:
• First session: we calculate the mass of each disk in order to calculate the total inertia of rotating disk.
• Second session: we figure out the angular acceleration of rotating disk by analyzing the video when the rotating disk rotates and stops by itself (of course under the effect of friction)
• Third session: we calculate the theoretical value of time, obtain the experimental value from an actual experiment. Compare two values to see how well we do the experiment.
• For the first session, the inertia of rotating disk is equal 0.019986 kg*m^2
• For the second session, the angular acceleration is equal -0.9048 rad/s^2
• For the last session, the percent error is 2.2%.
• There are some sort of uncertainties in this experiment:
• Uncertainty from the angle measured
• Uncertainty from the radii measured.
• Uncertainty from timing the time the cart takes to roll down a distance of 1 meter.
• Uncertainty in dotting points when analyzing the video to figure out angular acceleration.